Multi-Queue Switch Management in QoS Networks

1
Coordination Mechanisms for Unrelated Machine
Scheduling
Yossi Azar joint work with Kamal Jain Vahab
Mirrokni
2
Price on Anarchy KP, RT

• Selfish users
• User goal minimize its cost
• Nash Equilibrium (NE)
• System goal (e.g. Social welfare)
• The worst ratio of NE cost to OPT cost

3
Price of Anarchy Concept

• Not algorithmic
• Only analysis
• What to do if PoA is large
• How to influence the system

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Possible Solutions

• Change the system (add tolls, payments)
• Stackelberg strategy control some users
• Disadvantages changing the settings, global
  knowledge
• Challenge influence within the same setting and
  locally (distributed)

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Coordination Mechanism CKN

• Mechanism local policy (algorithm) that assigns
  a cost for each strategy of the user
• Advantages local, same type of cost
• Goal achieving good NE
• Example scheduling jobs on machines

6
Unrelated Machine Scheduling

• m unrelated machines
• n jobs each owned by different user
• p(i,j) - processing time of job i on machine j
• System goal minimize completion time
• User goal minimize its own completion time

• m unrelated machines
• n jobs each owned by different user
• p(i,j) - processing time of job i on machine j
• System goal minimize completion time
• User goal minimize its own completion time

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Unrelated Machines Scheduling
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Coordination Mechanism for Scheduling

• Policy for each machine (algorithm) which
• decides how to schedule jobs assigned to it
• Each Policy induces NE on jobs

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Local Scheduling Policies
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Type of Policies

• Local policy depends on jobs assigned to
  machine
• Strongly local policy - depends only on
  processing time of jobs on that machine
• Ordering Policy IIA (independence of irrelevant
  alternative)

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Challenge

• Design policies that results in
• good NE (i.e. low PoA)

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PoA of Longest First

• Results in poor NE
• The PoA is unbounded even for 2 machines
• The optimum completion time is low
• The completion time of NE is large

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Unrelated Machines Scheduling
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Equilibrium for Longest First
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Previous Results

• Identical Machines constant CKN
• Related constant, log m CV,ILMS
• Restricted assignment log m ILMS
• Unrelated Machines m (IK,DJ,ILMS)

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Main Results

• Negative Results (strongly local)
• PoA of any strongly local policy-at least m/2
• In particular, PoA of Shortest-First is of order
  m
• Resolve an open question from 1977
• (Alg D by Ibarra and Kim)

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Main Results

• Positive Results (local)
• Local ordering policy with PoA of O(log m)
• Any local ordering policy at least log m
• Pure Nash Convergence ? O(log2 m)
• More results on convergence

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Lower Bound for Strongly Local Policy

• We start with Shortest-First
• Extend it to arbitrary strongly local IIA policy
• Shortest-First is interesting by its own

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Shortest-First

• Approx factor known to be at most m
• NE can be computed by shortest-first greedy
  algorithm
• (Alg D by Ibarra and Kim)
• An open question from 1977
• We show it is at least m/2

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Idea of the Proof

• m types of jobs
• Type j can be scheduled on machines j j1
• Processing time of type j on machine j is low and
  on machine j1 is high (ratio is j)
• All jobs on machine j have almost the same
  processing time

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Example for Shortest-First
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Idea of the Proof

• OPT assign all jobs of type j to machine j
• Number of jobs is chosen such that OPT has the
  same completion time for all machines

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Optimal Assignment
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Idea of the Proof

• In NE about half jobs of type j are on machine j
  and half on machine j1
• Completion time of NE grows linearly in m

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Equilibrium for Shortest-First
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Extend to Arbitrary Strongly Local

• Structure is similar to lower bound for
  Shortest-First
• Arbitrary ordering function is given for each
  machine
• Indices of jobs are chosen to behave similar to
  the above example

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Efficiency Based Algorithm

• Order jobs on each machine by their efficiency
• Efficiency of job on machine is
• The ratio between jobs best processing time to
  its processing time on this machine
• PoA of algorithm is O(log m)

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Equilibrium Improves
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Efficiency Based Algorithm

• Unfortunately pure NE may not exist
• Iterative improvement may cycle
• Modified algorithm guarantees convergence and
  pure NE with PoA of O(log2 m)

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Modified Algorithm

• Each machine simulate log m submachines (by round
  robin)
• Submachine k of machine j handles jobs on
  efficiency between 2-k and 2-k1
• Jobs are ordered on submachine by Shortest-First
• PoA of algorithm is O(log2 m)

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Summary

• Coordination Mechanism
• Influence on the quality of the equilibrium
• Unrelated Machines
• m lower bound
• Shortest-First is at least m
• Local order by efficiency O(log m) optimal
• Pure Convergence O(log2 m)

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Discussion and Open Problems

• Non ordering strategies get below log m
• Extend to network routing
• Show more effective usage of coordination
  mechanism